Comments on A,B,C Chains of Heterotic and Type II Vacua

TL;DR

The paper probes F-theory/heterotic duality for six-dimensional vacua with non-semisimple gauge backgrounds by constructing Calabi–Yau hypersurfaces in toric varieties dual to $E_8\times E_8$ heterotic compactifications on $K3$. It extends prior work by systematically obtaining B and C chain models through extremal conifold transitions from A-models and demonstrates how ADE gauge enhancements and extra tensor multiplets arise within the toric framework. The authors show that the Dynkin diagrams of unbroken gauge groups are embedded in the polyhedra and map these toric constructions to heterotic data via Hodge numbers, confirming duality across a broader class of vacua. They also describe nonperturbative vacua with tensor multiplets obtained by base blow-ups and conifold transitions, highlighting a coherent toric strategy to organize and connect diverse heterotic/F-theory vacua.

Abstract

We construct, as hypersurfaces in toric varieties, Calabi-Yau manifolds corresponding to F-theory vacua dual to E8*E8 heterotic strings compactified to six dimensions on K3 surfaces with non-semisimple gauge backgrounds. These vacua were studied in the recent work of Aldazabal, Font, Ibanez and Uranga. We extend their results by constructing many more examples, corresponding to enhanced gauge symmetries, by noting that they can be obtained from previously known Calabi-Yau manifolds corresponding to K3 compactification of heterotic strings with simple gauge backgrounds by means of extremal transitions of the conifold type.